On infinity

Similarities between al-Kindī and David Hilbert On infinity

Al-Kindī aimed at a mathematical contradiction in the heart of the Aristotelian issue of actual infinity. In other words, actual infinity is self-canceling, and the only thing left is infinity on the potential level. Consider this example for the sake of simplification:

When we count from number 1 to number 2, we can count fractions in between such as: 1.1, 1.11, 1.12, 1.13, 1.14, . . . 1.2, . . . 1.3, . . . 1.9, 1.91, 1.92, . . . 1.99, 1.999, 1.9999, and we can count 1.9n as long as we can count, to infinity (without reaching it, of course), but no matter how many 1.99999n we can count, we know that the actual end is number 2, and we are only potentially going between 1 and 2, and what actually exists is the finite limit of number 1 and that is number 2. The counting of 1.99999n to infinity is not achievable and is actually impossible to reach; however, we can potentially assume that the process of counting could go on endlessly.

Al-Kindī’s notion of mathematical infinity is what revealed that Aristotelian cosmology is contradictory. Bodies that exist in actuality have a finite magnitude, and if we think about a body as infinite, that is only because it happens on the level of ideas but not in reality. Natural objects are necessarily finite.

Al-Kindī’s appeal to pure mathematics, by refuting this metaphysical notion, is an attempt to establish belief in God. His work in metaphysics might be considered one of his most significant contributions to Islamic philosophy. Aristotle mentioned the idea that actual infinity is impossible; while al-Kindī’s contribution was his ability to use an Aristotelian premise in order to go beyond Aristotelian cosmology itself and establish a totally different cosmos in which God is the creator. Al-Kindī also appeals to the notion of infinity in order to support religious belief by scientific knowledge, by assuming that the truth is one and could be reached by reason and revelation without contradiction. (Al-Ghazālī has a similar perspective and argument, as he explains in his book The incoherence of the philosophers [Tahāfut al-falāsifa], which we discuss later.)

Al-Kindī singles out the notion of infinity in order to delve into it mathematically, apply it to the natural sciences, and address it philosophically in his metaphysics. Since the time of al-Kindī, few philosophers or mathematicians addressed the notion of “infinity” in depth, until the twentieth century when a great mathematician, David Hilbert (1862–1943), published his work “On the infinite.”

Hilbert says, “[T]he definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but rather for the honor of the human understanding itself.”[1]

He discusses infinity in both fields, microphysics and macrophysics. After he mentions physics and the divisibility of atoms, particles, electrons, quanta, and how none of these permits infinite division in an absolute and unrestricted way, he says,

And the net result is, certainly, that we do not find anywhere in reality a homo-geneous continuum that permits of continued division and hence would realize the infinite in the small. The infinite divisibility of a continuum is an operation that is present only in our thoughts; it is merely an idea, which is refuted by our observation of nature and by the experience gained in physics and chemistry.[2]

The infinite cannot actually be found “anywhere in reality,” especially in physics. It exists only on the level of potentiality, “in our thoughts”; it is an idea. A physical object that exists outside the mind is finite. The body of the world that exists outside is finite, and thus it cannot be eternal. At the end of his paper Hilbert says: “The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking—a remarkable harmony between being and thought.”[3]

[1] David Hilbert, “On the infinite,” in From Frege to Godel: A source book in mathematical logic, 1879–1931, ed. Jean van Heijenoort (Cambridge, MA: Harvard University Press, 1967), pp. 370–371.